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\title{高等代数一}
\subtitle{21-习题与问答-子空间的基-向量组的秩 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年12月6日} }

\maketitle

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\begin{enumerate}

\item  子空间的交空间与和空间
\item  线性相关的向量组、线性无关的向量组
\item  向量组的极大线性无关组
\item  向量组的秩
\item  向量空间的基
\item  向量空间的维数

\end{enumerate}


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{\small 
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
4-习题&8-习题&12-习题&16-习题&21-习题&25-习题 \\ \hline 
{01}&{02}&03&04&\underline{05}&06 \\   
{07}&{08}&09&10&\underline{11}&12 \\  
{13}&{14}&15&16&\underline{17}&18 \\ 
{19}&{20}&21&22&\underline{23}&24 \\  
{25}&{26}&27&28&\underline{29}&30 \\  
{31}&{32}&33&34&\underline{35}&36 \\  
{37}&{38}&39&40&\underline{41}&42 \\  
{43}&{44}&45&46&\underline{47}&48 \\ 
{49}&{50}&51&52&\underline{53}&54 \\  
\end{tabular}
\end{table}
}

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\begin{itemize}

\item  习题1：设 $V$ 是一个向量空间。设 $W_1$ 与 $W_2$ 是 $V$ 的两个向量子空间，且 $W_1\subsetneq V$, $W_2\subsetneq V$. 证明 $W_1\cup W_2 \subsetneq V$. 

\item  解答思路：分情况讨论。
\begin{enumerate}
\item  当 $W_1\subseteq W_2$ 或 $W_2\subseteq W_1$ 时。
\item  当  $W_1\nsubseteq W_2$ 且 $W_2\nsubseteq W_1$ 时。这时从 $W_1$ 与 $W_2$ 中各取一个元素，加起来。
\end{enumerate}

\end{itemize}

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\begin{itemize}

\item  习题2：设 $V$ 是一个向量空间。设 $W,W_1,W_2$ 都是 $V$ 的子空间。设 $W_1\subseteq W_2$, $W+W_1=W+W_2$ 且 $W\cap W_1=W\cap W_2$. 证明 $W_1=W_2$. 

\item  解答思路：设 $\alpha\in W_2$. 由 $W+W_1=W+W_2$ 得 $\alpha\in W+W_1$. 


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\begin{itemize}

\item  习题3：设 $\alpha,\beta, \gamma, \delta$ 是下述矩阵 $A$ 的列向量，
\begin{eqnarray*}
A=\begin{pmatrix} a_1&b_1&c_1&d_1 \\  a_2&b_2&c_2&d_2 \\ a_3&b_3&c_3&d_3 \\ a_4&b_4&c_4&d_4  \end{pmatrix}
= ( \alpha,\beta, \gamma, \delta ). 
\end{eqnarray*}
设 $\det(A)\neq 0$. 证明向量组 $\{\alpha,\beta, \gamma, \delta\}$ 线性无关。

\item  解答思路：按线性无关的定义验证即可。


\end{itemize}

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\begin{itemize}

\item  习题4：设有$V_1=\mathbb{R}^4$ 与 $V_2=\mathbb{R}^5$ 中的向量组
\begin{eqnarray*}
\left\{\begin{array}{l}
\alpha_1=(a_1,b_1,c_1,d_1), \\ 
\alpha_2=(a_2,b_2,c_2,d_2), \\ 
\alpha_3=(a_3,b_3,c_3,d_3), \\ 
\end{array}\right. 
\hspace{0.3cm}
\left\{\begin{array}{l}
\beta_1=(a_1,b_1,c_1,d_1,e_1), \\ 
\beta_2=(a_2,b_2,c_2,d_2,e_2), \\ 
\beta_3=(a_3,b_3,c_3,d_3,e_3).  \\ 
\end{array}\right. 
\end{eqnarray*}
设向量组 $\{\alpha_1,\alpha_2,\alpha_3\}$ 线性无关。证明向量组 $\{\beta_1,\beta_2,\beta_3\}$ 也线性无关。

\item  解答思路：按线性无关的定义验证。

\end{itemize}

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\begin{itemize}

\item  习题5：设向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性无关。设 
\begin{eqnarray*}
\left\{\begin{array}{l}
\beta_1=\alpha_1+\alpha_r, \\ 
\beta_2=\alpha_2+\alpha_r, \\ 
\vdots \\ 
\beta_{r-1}=\alpha_{r-1}+\alpha_r, \\ 
\beta_r=\alpha_r. 
\end{array}\right. 
\end{eqnarray*}
证明向量组 $\{\beta_1, \beta_2, \cdots, \beta_r\}$ 也线性无关。

\item  解答思路：按线性无关的定义验证。


\end{itemize}

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\begin{itemize}

\item  习题6：判断正误，并说明理由，
\begin{enumerate}
\item[A.]  如果当 $k_1=k_2=\cdots=k_r=0$ 时，有 $k_1\alpha_1+k_2\alpha_2+\cdots+k_r\alpha_r=\theta$, 
那么向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性无关。
\item[B.]  如果向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性无关，而且向量 $\beta$ 不能由向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性表示，那么向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r, \beta\}$ 线性无关。
\item[C.]  如果向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性无关，那么其中每一个向量都不是其余向量的线性组合。
\item[D.]  如果向量组 $\{\alpha_1, \alpha_2, \cdots, \alpha_r\}$ 线性相关，那么其中每一个向量都是其余向量的线性组合。
\end{enumerate}

\item  答案：AD错误，BC正确。


\end{itemize}

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\begin{itemize}

\item  习题7：设向量 $\beta$ 可由向量组 $\{\alpha_1,\alpha_2,\alpha_3,\gamma\}$ 线性表示，但不能由向量组 $\{\alpha_1,\alpha_2,\alpha_3\}$ 线性表示。证明向量组 $\{\alpha_1,\alpha_2,\alpha_3,\gamma\}$ 与向量组 $\{\alpha_1,\alpha_2,\alpha_3,\beta\}$ 等价。

\item  解答思路：按等价的向量组的定义验证。仅需验证向量 $\gamma$ 可由向量组 $\{\alpha_1,\alpha_2,\alpha_3,\beta\}$ 线性表示。


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\begin{itemize}

\item  习题8：设向量空间 $V=\mathbb{R}^4$ 中的一些向量如下，
\begin{eqnarray*}
%\left\{\begin{array}{l}
\alpha_1=(2,-3,4,2), \,\,
\alpha_2=(1,4,-2,-1), \,\,
\alpha_3=(3,-2,4,1), \,\, 
\alpha_4=(2,0,2,2). 
%\end{array}\right. 
\end{eqnarray*}
设 $W_1=L(\alpha_1,\alpha_2)$, $W_2=L(\alpha_3,\alpha_4)$. 求 $W_1+W_2$ 与 $W_1\cap W_2$ 的维数。

\item  答案：$W_1$ 与 $W_2$ 的和空间与交空间的维数分别是 3 与 1. 


\end{itemize}

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\begin{itemize}

\item  习题9：求下述子空间的维数，
\begin{enumerate}
\item  $W=L( 1+x, (1+x)^2, (1+x)^3, 1-x, (1-x)^2, (1-x)^3) \subseteq \mathbb{R}[x]$. 
\item  $W=L(1, \sin(x), \sin(2x), \cos(x), \cos(2x)) \subseteq C[0,2\pi]$. 
\end{enumerate}

\item  解答思路：
\begin{enumerate}
\item  首先建立向量空间 $\mathbb{R}[x]_3$ 与 $\mathbb{R}^4$ 的一个同构。维数为4. 
\item  画出这些函数的图像。判断这个向量组是否线性相关。维数为5. 
\end{enumerate}

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\begin{itemize}

\item  习题10：将下述向量组扩充为 $\mathbb{R}^4$ 的一个基， $$\{\alpha_1=(2,0,2,2), \alpha_2=(2,0,2,3)\}. $$

\item  解答思路：一种方法是从下述标准基中选取两个向量， $$\{\varepsilon_1=(1,0,0,0), \varepsilon_2=(0,1,0,0), \varepsilon_3=(0,0,1,0), \varepsilon_4=(0,0,0,1) \}. $$


\end{itemize}

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